For a familiar subset of the friendly numbers, then YES, there is a proven relationship between that subset and triangular numbers.
It doesn't seem to be well-emphasized by Mathworld and Wikipedia that every even perfect number is also a friendly number. To recall, a friendly number is one of integers $(m,n)$ such that,
$$k=\frac{\sigma(m)}m =\frac{\sigma(n)}n$$
where $\sigma(n)$ is the divisor function and $k$ is some rational. However, when $k$ is an integer, then we have the multiperfect numbers of class $k$. For $k=2$, these are just the perfect numbers,
$$2 =\frac{\sigma(6)}6 = \frac{\sigma(28)}{28} = \frac{\sigma(496)}{496} = \frac{\sigma(8128)}{8128} = \dots$$
So every even perfect numbers are automatically friendly numbers. And it is long known that the even perfect number $P_p$ also has triangular form $\frac{n(n+1)}2$,
$$P_p = \frac{M_p(M_p+1)}2$$
where $M_p$ is a Mersenne prime.
Conclusion: There is a proven relationship between "friendly numbers" and "triangular numbers" whenever it is also an "even perfect number".
