Given a line $ax+by=c$ where $a,b,c$ are positive integers. Is there any formula to find the number of points inside the triangle formed by this line, $x$-axis and $y$-axis? Points on the boundary counts too.
Pick's theorem doesn't work because axis intersections are not necessarily lattice points.
For any such point $(x,y)$ from the inequality $ax+by\le c$ it follows that the number is $$\sum_{k=0}^{\lfloor C/A\rfloor}\left\lfloor\frac{C-Ak}{B}\right\rfloor=\sum_{k=0}^{\lfloor C/A\rfloor}\frac{C-Ak}{B}-\sum_{k=0}^{\lfloor C/A\rfloor}\left\{\frac{C-Ak}{B}\right\}$$ but I don't know how to proceed from here.
The extension of Pick's theorem you are looking for may be Ehrart theory https://en.wikipedia.org/wiki/Ehrhart_polynomial because one of its extensions deals with vertices having rational coordinates (this is the case here because the lines' coefficients are integers).