No they will not do it for 'free'. They are in it to make money, or CYA. So if you brought in your GPU and RECEIPT (because you bought it from them) and want them to check the GPU is working, yeah they can do that, but otherwise NO.
Your only choice is to BUY a new PSU and try that out first. Always good to have a spare or upgrade anyway. Worst case if you buy it like from Bestbuy and it didn't "fix" anything you can bring it back and ask for a refund as you 'don't need' the PSU, it wasn't the problem with your PC.