Is it possible to generate a 2D distribution function $f(x,y)$ with function supports specified as $[-a,a]$ and $[-b,b]$ for $x$ and $y$ respectively, such that it always has moments which are NON ZERO up-to a certain higher order say $p+q$?
Every distribution whose suppport is bounded has moments of every order. Here, the support is included in $[-a,a]\times[-b,b]$ hence bounded.
If the support of the distribution is included in $[a,b]\times[c,d]$ with $0\lt a\lt b$ and $0\lt c\lt d$, then the $(i,j)$-moment is between $a^ic^j$ and $b^id^j$ for every nonnegative $i$ and $j$, in particular this moment is finite and positive.
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You could use the Moment Generating Function? Up to some conditions, i.e. when it exists, the MGF uniquely defines a distribution, by basically defining all its moments.