Since $\int_1^\infty f(x)\,dx$ and $\int_1^\infty g(x) \,dx$ are converge, $\lim_{x\to\infty}f(x)=0$ and $\lim_{x\to\infty}g(x)=0$, so $f,g$ are fractional function, and denominator is changing more fast than numerator.
The product of two functions has the same situation (i.e. denominator is changing more fast than numerator)
So how can we find such functions?
Let $f(x)=g(x)=\frac{\sin(x)}{\sqrt x}$. Then, clearly we see that $\int_1^\infty f(x)\,dx=\int_1^\infty g(x)\,dx$ are convergent.
But, $\int_1^\infty f(x)g(x)\,dx=\int_1^\infty \frac{\sin^2(x)}{x}\,dx$ diverges.
If you have hard time finding problems near $\infty$, find them near $1$. Take $f=g$ and $f(x) = \frac{1}{\sqrt{x-1}} $ for $x \in (1,2]$ and $f(x) = \frac{1}{x^2 - 3}$ for $x \in (2,+\infty)$. Clearly $f$ is continuous and the integral is convergent, but $fg$ near $1$ is of form $\frac{1}{x-1}$, which will diverge.
The simplest example is perhaps $f(x)=g(x)=\frac 1 {\sqrt {x-1}}$. (Unless you want the functions defined and continuous at $1$ also).
