solutions of consecutive integers  and observations I wonder that, I made these observations from my previous study on product of consecutive integers. I am looking the solutions of these kind of equations. 
$(1)$ Is $x(x+1)(x+2)...(x+[\text{any-odd-integer}]) = y^2$ has solutions or not?. If exits, how to list them?
$(2)$Prove that  For  any $k \ne 2,4$,a polynomial function of a form $x(x+1)(x+2)...(x+k-1)+Q = t^2$, where Q is a rational number has solutions or not? here $x$ is variable and others maybe constants. Your answer is very essential for me.
 A: Please note that this question has already been posted here on Math.stackexchange.
You seem to be trying to extend the result from this paper: "On Diophantine equations of the form $(x−a1)(x−a_2)…(x−a_k)+r=y^n.$" by Manisha Kulkarni and B.Sury, which is available here to the case $n=2$.
(This should probably have been a comment, but I don't think I have enough reputation)
A: 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.  
