The question is: By how much seconds does it need to increase?
I would be very hesitant with allowing a very scarce resource to be over-utilized by both anonymous and registered users. I strongly believe increasing the allowed run-time has the potential to make the overall experience worse for all users.
A badly designed query can not be salvaged by throwing more CPU and IO at it. A badly designed query needs a better execution plan (prefer seek over scan). I'm told storage space is not much of a concern. As such adding an index to help a slow query is a better approach because that index is likely to help more queries run with less resources. And there is a lot to win as shown by my previous request: Can we exchange one index for another one?
If you aren't able to analyze and/or improve a query on your own you can ask for query improvements in SEDE chat, on Meta or even on Database Administrators or Code Review. The latter two sites expect more effort on your end with expert knowledge in return.