Is there a general method or formula for calculating the infinite sum $$\sum_{n=1}^{\infty} 1/(an^{2}+bn+c)$$?
It depends what you call "calculating". If you want the result in "closed form", any series of rational functions of $$n$$ can be summed in terms of the logarithmic derivative $$\psi$$ of the gamma function and its derivatives, simply by decomposing into partial fractions. If you want a numerical approximation, you can either use this, or the many summation methods existing in the literature which can give you thousands of decimals in fractions of a second.