no.. in LHS: f and g is inside the integral sign. in RHS, f will be outside of the integral sign.
these are the some properties (as far as i know) that the convolution function have:
symmetric: f*g=g*f
associative (in convolution): (f*g)*h=f*(g*h)
distributive over addition: f*(g+h) = f*g + f*h
associative and symmetric in real multiplication: a(f*g)=(af)*g=f*(ag)
also: (f*g)=(Df)*g = f* (Dg)