Would a long awaited Anfield win and progression to an FA Cup final see fans jump back on the 'give Martinez more time' boat again? I've got a feeling it would.
If by some miraculous miracle this scenario was to happen, personally I'd still want him gone after the final anyway.
Taking your post to mean WINNING the cup, rather than just getting to the final and losing... my answer is no.
The reason for this is as follows:
We are currently 12th. We have 38 points, exactly the same as Bournemouth, whom we are ahead of only due to goal difference. We have two games in hand over them. In 14th and 15th, on 37 points each, are Watford and Swansea. Palace are 16th, with 34 points. Ahead of us in 11th is WBA, on 40 points. In 10th place with 44 points is Chelsea, and the RS are a point ahead of them, sitting in 9th with 45 points.
What does all this mean? It means that MATHEMATICALLY there is as much chance of us coming 16th at this point as there is of us coming 9th. 17th placed Norwich are closer to catching us than we are to catching Stoke (8th) or Southampton (7th). Yes, we have games in hand over everyone around us, but I have serious concerns over our chances of picking up decent points from our remaining games. We face Watford and Palace away in our next two PL fixtures, followed by Southampton at Goodison (where no doubt the fans will be blamed if the win doesn't materialise) and then the derby at the pit. We finish off with Bournemouth, Leicester, Sunderland and Norwich.
Some of those teams will be scrapping for their PL lives when we play them, Leicester will be trying to win the title, Southampton are looking at an EL place and the RS need no motivation when we play each other. Our opponents will ALL be after three points when they face us, we have no "easy" games left.
You asked if winning the Derby and winning the FA cup final would get fans back onside - I would say that if we won those games but lost all the other PL fixtures, which at this point looks somewhat possible, we'd be 16th or 17th. Sunderland and Norwich aren't going to go on six-game winning streaks now, so we are safe from relegation, but 16th place will see a mass exodus of talent form the club. Say we scrape a win or two, and a couple draws, and finish 14th - is 14th place a price you are willing to pay for a Cup? Not me.
Martinez needs to win the majority (i.e. at LEAST 5 of the 8 remaining PL games and ALL the FA cup fixtures) of his remaining games to win back my support. Those wins MUST include the Derby and the FA Cup final, and there must be enough of them to finish higher than at least one of either Chelsea or the RS.
10th place, the FA cup, and finishing higher than just ONE of the big-spending clubs. Anything less than that and my opinion stays as it is right now: sack him.