By quads I mean four sided shapes. You can add vertex anywhere inside the triangle, but you can not add vertex onto existing edges, i.e., splitting them.
I tried but currently it appears to be impossible (see figure below for an attempt). If I guess correctly there must be some topological constraints that make this impossible.
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Nah, it's impossible. Let's say it were possible, and you had a working dissection. Now imagine you draw four points in each quadrilateral, one on each side. Now of course the number of points is divisible by four, so it is even. On the other hand, you can pair all but three points up with the corresponding point on the other edge, so the number of points is three more than an even number, therefore it must be odd as well.
This is a contradiction, hence the dissection you are looking for is impossible.
Let's assume you could divide everything into quadrilaterals. Now, let's count the number of edges in this division; call this number $E$ (in your example, I count $E=15$, but I might have miscounted). Now, each quadrilateral has 4 edges, but with the exception of the 3 edges on the border of the triangle, each edge belongs to 2 quadrilaterals. Therefore, if there are $Q$ quadrilaterals, we have that $4Q = 2E-3$.
But the left side of this equation is even and the right side is odd, leading to a contradiction.
In general, a similar argument will show that you always need an even number of odd-sided faces in any planar graph.