Of course Aston villa will always want to fight on behalf of liverpool since psg is the one that got them Eliminated in the champions league, definitely Aston villa wouldn't want to be Eliminated as well by psg this is going to make it very tough for psg. and if they're not careful I won't be surprise to see Aston villa defeating them in the champions league because Aston villa is not that easy to deal with.
There is no reason like this with Aston Villa, and they are not fan or supporter of Liverpool. They fight with PSG for their own ticket to semi final phase of Champions League, to make a new history of their club, that's it. Nothing to force them fighting on behalf of Liverpool, nothing.
Aston Villa don't play too good in Premier League this season but their adventure in Champions League reminding me about Porto, Monaco, Napoli and Atalanta, generally underdogs in past seasons. Aston Villa even have worse player resource and quality than those underdogs and they will make very big surprise if beating PSG in quarter finals and be eligible for semi finals. It will be a first time their club can participate in UEFA Champions League semi finals so it's worthy to fight.