Answer: 45

Solution:

Let $k = \frac{AM}{CM}.$ Triangles $ABM$ and $CBM$ share the altitude from $B,$ so $\frac{[ABM]}{[CBM]} = k.$ Since the inradius of a triangle is its area divided by its semiperimeter, equal inradii force $\frac{12 + AM + BM}{13 + CM + BM} = k$ as well. From $AM + CM = 15$ we get $AM = \frac{15k}{k+1}$ and $CM = \frac{15}{k+1};$ since $AM = k \cdot CM,$ the perimeter equation simplifies to $BM(1 - k) = 13k - 12,$ so $$BM = \frac{13k - 12}{1 - k},$$ and $BM \gt 0$ forces $\frac{12}{13} \lt k \lt 1.$

Stewart's theorem on cevian $\overline{BM}$ gives $AB^2 \cdot CM + BC^2 \cdot AM$ $= AC\left(BM^2 + AM \cdot CM\right),$ so $$\begin{aligned} BM^2 &= \frac{144 + 169k}{k + 1} \\ &\quad {}- \frac{225k}{(k+1)^2}. \end{aligned}$$ Setting this equal to $\frac{(13k-12)^2}{(1-k)^2}$ and clearing denominators yields $(169k^2 + 88k + 144)(1-k)^2$ $= (13k-12)^2(k+1)^2,$ which simplifies to $4k\left(69k^2 - 112k + 44\right) = 0.$

The roots are $k = 0,$ $k = \frac{2}{3},$ and $k = \frac{22}{23},$ and only $k = \frac{22}{23}$ exceeds $\frac{12}{13}$ (then $AM = \frac{22}{3},$ $CM = \frac{23}{3},$ $BM = 10$). Hence $p + q = 22 + 23 = 45.$