Here's another one. a little strange, since the quadrature rule involve the value of line integrals, but I guess you can write those using a separate quadrature rule...
Bojanov, Borislav; Petrova, Guergana, Numerical integration over a disc. A new Gaussian quadrature formula, Numer. Math. 80, No. 1, 39–59 (1998). ZBL0911.65015.
We construct a quadrature formula for integration on the unit disc which is based on line integrals over $n$ distinct chords in the disc and integrates exactly all polynomials in two variables of total degree $2n - 1$.
Cheers.
