I gather you mean the MathieuC function? [Or the MathieuCE?]. I will assume MathieuC.

There certainly does not seem to be anything obvious in the "standard" references, nor a quick Google away. However, the (algebraic) version of the Mathieu function is a special case of the HeunC function. And there are interesting integral representations for those, for example in this paper (see p.9 for example). Yes, those representations are in terms of HeunC itself -- however observe the ``free parameter'' c introduced in equation (6.6). The idea then would be to pick a parameter value for this c that causes the HeunC function to "collapse" to something simpler, thus giving you an honest-to-goodness integral transform.

The other trick is to try to find good kernel functions. Lemma 3.3 from that paper allows quite a bit of flexibility in that regard.

Unfortunately, the obvious representations do not help: both Laplace transforms and Fourier transforms end up giving an ODE whose solution is again in terms of Mathieu functions.