Regarding you first question: no this does not have any solution by a result of Erdős "Note on product of consecutive integers" (Journal London Math. Soc. 1939); you do not need the condition 'odd' either.

I do not understand the second question. For some $Q$ of course there will be some solution, for others not;
but in general it is unclear to me what quantities are the variables and which are to be considered constant.

*Added* (in view of the other answer/comment): There are various investigations around this; a somewhat recent paper that seems still closer to this question would be Bilu, Kulkarni, Sury "The Diophantine equation $x(x+1) \dots (x+ (m-1))+ r=y^n$" (note the $n$ can be $2$).

A result from there implies that (coming back to the question, so $n=2$) if $r$ is not a square, then there are only finitely many solutions (and these can be determined explicitly).
And other results are mentioned. In general, there is quite a bit of literature on this type of question. Without further information from OP it is not really clear what type of information would be relevant.