Take $a=1/4$.
Maple says
$$
{}_1F_2\left(1;\frac14,\frac34;-x^2\right) =
1-2\,\sqrt {x}\sqrt {\pi}\sin \left( 2\,x \right)\, {C}\!
\left( 2\,{\frac {\sqrt {x}}{\sqrt {\pi}}} \right) +2\,\sqrt {x}
\sqrt {\pi}\cos \left( 2\,x \right)\, {S}\! \left( 2\,{\frac {
\sqrt {x}}{\sqrt {\pi}}} \right)
$$
in terms of the Fresnel integrals $S$ and $C$.
Now, $S$ and $C$ are not elementary. So I guess this combination is also not elementary.
Simplified ... $$ {}_1F_2\left(1;\frac14,\frac34;-x^2\right) = 1+2\sqrt {x}\int_{-x}^{0}\!{\frac {\sin \left( 2\,r \right) }{\sqrt {r+x}}}\,{\rm d}r,\qquad x>0. $$ The proof that $S$ is not elementary may also work for this?