Does this equation hold in a bilinear map? I would like to verify whether or not the following equation holds:

$e(a,c)^{c1\cdot c2\cdot c3}e(b,c)^{c1\cdot c2\cdot c4}==e(a,c)^{c2\cdot c3}e(b,c)^{c1^2\cdot c2\cdot c4}$ for appropriately defined bilinear map $e$ with different input groups $G_1$ and $G_2$ for $a,b\in G_1$ and $c\in G_2$
No it does not. To see it, take $b=1$:

$e(a,c)^{c1\cdot c2\cdot c3}e(b,c)^{c1\cdot c2\cdot c4}= e(a,c)^{c1\cdot c2\cdot c3}e(1,c)^{c1\cdot c2\cdot c4} = e(a,c)^{c1\cdot c2\cdot c3}1^{c1\cdot c2\cdot c4} = e(a,c)^{c1\cdot c2\cdot c3}$

similarly,

$e(a,c)^{c2\cdot c3}e(b,c)^{c1^2\cdot c2\cdot c4} = e(a,c)^{c2\cdot c3}$

So you would get

$e(a,c)^{c1\cdot c2\cdot c3} = e(a,c)^{c2\cdot c3}$

which is clearly not true in general (unless $c_1 = 1$ or $c_1 = 0$ or $e(a,c) = 1$).