Can I marry Aela the huntress after completing the quest: Liberate Skyrim?
Answer to this is Probably Not
Aela has a strong loyalty to Jorrvaskr. Which has its loyalty to the Imperials. If the Imperials win then Vignar Gray-Mane is the new Jarl of Whiterun.
She is part of many factions but these are of importance :
- GuardFaction (makes an actor a guard)
- PotentialMarriageFaction
- TownWhiterunFaction
She is part of Town Whiterun faction and Guard faction. Which automatically makes her not available for marriage in your case, since entire of Whiterun will not have Whiterun guards. They are replaced by Stormcloak soldiers.
Also according to her entry at UESP:
Aela is part of the Guard Faction. There is a bug wherein due to being part of the Guard faction, Aela may behave erratically if she's following you when beginning the Battle For Whiterun quest.
According to entry for Battle For Whiterun at UESP Stormcloak version:
Also during Battle For Whiterun (Stormcloak version), Aela the Huntress may disappear if she is your wife at the time. She may come back eventually after you moved on to the next Quest, Liberation of Skyrim.
Since you have completed Liberation of Skyrim quest and still can not find Aela, you probably might not be able to marry her.