Can a Goertzel filter response be mapped to a Direct Form II Biquad IIR Filter
try it, you'll like it.  

the form in the Jocobson &amp; Lyons paper is Direct Form II, with the coefficients explicit.  the form in the CMSIS source is a transpose of a Direct Form I, i believe, and the one in the Lyons &amp; Bell paper is a Direct Form II (not transposed, but set $a_0=1$ because they should not have had that $a_0$ gain block in the path.).  the $b_0, b_1, b_2$ coefs come from the numerator and the $a_1, a_2$ come from the denominator and need to have their signs changed (since the scaled feedback signals are added and not subtracted in the feedback).

are you floating-point or fixed-point?  if the latter, the untransposed Direct Form I will serve you better.