Continuous-time frequency response question
Is it possible to express $X_c\left(j\Omega \frac{T_2}{T_1}\right)$ in terms of $X_c\left(j\Omega \right)$ using some continuous-time filter?


an LTI system cannot do that.  continuous-time "filters" need not be LTI but usually we mean LTI when we consider "filters" in analog or digital filters.

you can change the spectrum of $X_c\left(j\Omega \right)$ to $X_c\left(j\Omega \frac{T_2}{T_1}\right)$ by speeding up (or slowing down) the time index of $x_c(t)$ to $x_c\left(t \frac{T_1}{T_2}\right)$.

it appears you want a pitch shifter or similar.  is that it?