While both the former and the latter are entirely plausible things for Quirrell to do, it is also worth noting that Quirrell would happily play the Role of a concerned tutor before McGonagall at this time. It would make her trust him more at a time when he may need to use her and other teachers on short notice to fulfil his own objectives, even if he doesn’t intend to do anything about the Restricted Session at all.
While both the former and the latter are entirely plausible things for Quirrell to do, it is also worth noting that Quirrell would happily play the Role of a concerned tutor before McGonagall at this time. It would make her trust him more at a time when he may need to use her and other teachers on short notice to fulfil his own objectives, even if he doesn’t intend to do anything about the Restricted Session at all.