Bessel filter second-order sections Q and Fc multiplier derivation A practical guide for digital IIR audio filters has cookbook-style values for creating higher-order Bessel filters out of biquads, but the values listed aren't very precise:

You should multiply the Fc for each stage by the following coefficients:
1: 1.00
2: 1.27
3: 1.32 1.45
4: 1.60 1.43
5: 1.50 1.76 1.56
6: 1.90 1.69 1.60

The corresponding Q values are:
1: ---
2: 0.58
3: --- 0.69
4: 0.81 0.52
5: ---- 0.92 0.56
6: 1.02 0.61 0.51


What are the expressions to calculate these values exactly?

For the Butterworth table, the values are given by Q = 1/(2 * sin((pi / N) * (n + 1 / 2))), for instance.  So where the table says 4: 0.54 1.31, the equation gives 1.3065629648763766 and 0.54119610014619701.  But I can't find the expression for Bessel filters, except that 2: 0.58 is $1\over\sqrt3$.

Prototype-generating functions just have long tables of numbers, so maybe these are not easy to calculate?  (Bessel functions?)  If so, a listing of these Q and Fc values for the first several orders would be good.

(And yes, I know the bilinear transformed Bessel is no longer linear-phase.  :/  I just want to know where these numbers are from.)
There is no simple closed-form equation to calculate Bessel filter bi-quad coefficients. The poles of the filter come from a Bessel polynomial. Higher order Bessel polynomials are determined using a recursion relationship. You need computer algorithms to calculate the poles and resolve the bi-quad coefficients. 

Two books I can recommend: “Passive and Active Filters” by Wai-Kai Chen and “Analog Filter Design” by M.E. Van Valkenburg.