You get the quest "In My Time of Need" inside Whiterun, in the main gate, near the guard barracks. Most likely you will get it when you return from slaying the dragon near the western watchtower.

You see the guards arguing with a group of outsiders. The guards tell them that they are not allowed in Whiterun, and to go back to where they came from. The Alik'r warrior will tell the guards that they are not causing any trouble. All they are asking is for the guards to look for a certain person. The guard says he does not care what they are doing and that they are lucky he did not toss them in jail after what happened, and tells them to get lost. The Alik'r warrior will tell the guard that this is not over, they will be back

The Alik'r warrior turns to you now. They are looking for someone in Whiterun, and will pay good money for information. Ask him who they are looking for. He will tell you they are looking for a woman, a Redguard, a foreigner in these lands, like them. She is likely not using her true name. They will pay for any information regarding her location. They are not welcome in Whiterun, so they will be in Rorikstead should you learn anything. Ask them why they are looking for this person. They will tell you it's none of your concern, and all you need to know is that they are paying for information, and that you can walk away if that doesn't interest you. Ask them who they are looking for again. He will tell you that they are looking for a fugitive who comes from Hammerfell, A redguard woman. She may be somewhere in this city.

Open your quest journal.

In My Time of Need
A group of men calling themselves the Alik'r are looking for a Redguard woman in Whiterun

Find the Redguard woman.

Go to the Bannered Mare, near the market. Talk to Saadia there. Ask her if she knows that some Alik'r warriors are looking for a Redguard woman. She will be alarmed that they have found her and plead for help. She asks you to come talk to her in private. You can choose to either speak with her, or inform the Alik'r of Saadia's location.

If you want to go speak to her, follow her to her room upstairs. There she will draw a knife at you and threaten you not to touch her. Tell her they just asked you to find her. She says you cannot tell them. She needs your help. There isn't anyone she can trust in the city. Ask her what she wants. She will tell you she is not the person the people in whiterun think she is. Her name is Iman. She is a noble from Hammerfell. She wants you to drive the Alik'r away before they find her and execute her.Ask her how you are supposed to get rid of them. She will tell you that they are mercenaries led by Kematu and that if you kill their leader, the rest will scatter.

Ask her how you are supposed to find him. She will tell you that she heard one of his men got arrested trying to break into the city. He might know the whereabouts of the mercenary leader. Your quest journal is updated:

The Barmaid at the Bannered mare in Whiterun is apparently a fugitive noble from Hammerfell. She claims that she is on the run from assassins sent by rival houses in her homeland and needs someone to stop them before they find her. Talk to the Alik'r prisoner or inform the Alik'r of Saadia's location.

Head to Dragonsreach (You can fast travel there if you've already been there). Enter, then go to the Northwest and climb down some stairs to enter the Dragonsreach dungeon..

Find the cell where the prisoner is located and talk to him through the bars. Tell him you need to find Kematu. Ask him where Kematu is hiding. He will tell you that he will give you the information you need if you see to his release. Talk to the guard and pay thee 100g fine, then talk to the Alik'r prisoner again about how you two can help each other out. Tell him his fine has been paid and to tell you where Kematu is. He will tell you that Kematu is west of Whiterun, in an unassuming little cave called "Swindler's Den". Leave the dungeon and go back up to Dragonsreach. Your objective now is to kill Kematu.

Exit Whiterun and head west. If you have already discovered the Western Watchtower, you can save some time by fast travelling there. Work your way westward, past Fort Greymoor, and around the mountains, to the Swindler's Den. The cave entrance is located close to a  river. Kill the lone bandit that is guarding it and go inside.

Work your way through the tunnels, killing the bandits along the way. Reach a big cavern full of more bandits. The best strategy to kill those is to lure them into the narrow tunnel where you can kill them one at a time.

After defeating them, jump up to the upper level of the cavern and continue east through some water. Kematu will be in the next cavern. He will tell you that you have proven your strength and he wants to avoid further bldoodshed. You can choose to talk to Kematu or kill him. If you talk to him,  ask him why he is after Saadia. He will tell you she sold the city out to the Dominion, she is a traitor. She betrayed her city to the Dominipn and they want to bring her to justice. Tell him you were sent to kill him but he asks that instead you help with bringing her to Justice in Hammerfell.

He tells you that since she trusts you, Lure her to the stables outside Whiterun by telling her they are coming for her and she needs to leave. In the stables, they will be waiting for her and take her in. He will share a portion of the bounty with you. Tell him you will do it and he says they will wait for you at the stables:

"i've been asked to eliminate claims that saadia betrayed her people to the Dominion and they are here to bring her justice. I've agreed to lead her into an ambush at the whiterun stables"

Your objective is to lead Saadia to the Whiterun Stables

Leave Swinder's Den and fast travel to whiterun. Find Saadia at the Bannered Mare inn. If you can't find her in the ground level, then she is in her room on the second floor. Lie to her, that you were unable to defeat them all and that they are coming for her. They found a way into the city. You need to get her out of here. Tell her you have a horse waiting for her in the stables, and that you will make sure that she is safe. She trusts you. Exit the inn and lead her to the stables outside the city. Take her to the back of the stables. Kematu awaits her there. She asks you what have you done. Kematu tells her that he she couldnt expect to manipulate people for much longer, and paralyzes her with a spell.

Talk to Kematu. He will tell you that they are now going to take Saadia back to Hammerfell where she will pay a price for her treason. He will give you 500 gold as reward.

This completes the quest "In My Time of Need"