Measuring exact dielectric constant without equipment (after fabrication)

I want to ask how feasible it is in following cases:

Dielectric constant ~4, frequency 10 to 20GHz
Er used in design = 4, in reality 3.8

1) After fabrication of IQ receiver with wrong Er=4.
provide test signal to reciever, calculate phase error (IQ must be +/-90 for two test signals). Correct layout according to calculated phase differences. Is it common method?

2) Fabrication of lambda/4 stub test circuit, use it's output as LO for receiver. By adding/removing copper to lambda/4 stub find the length, at wich receiver stops operating (too low LO level indicates, that lambda/4 stub have correct length)

Without the equipment you cannot MEASURE the substrate permittivity. As your concern is making a quarter-wave stub, you can tune it by trimming its length or adding a capacitance at its live end.
Trimming is often used is microstrip circuits.

If you are using low cost FR4 (Er~4) in frequency range 10GHz to 20GHz, Er will be all over the values on the same PCB. All the microstrip theories would not apply in this situation.
For those microwave frequencies you might need to use high quality laminates (from Rogers, Taconic, etc) with low Er (between 2 and 3)

There is interesting paper here:
amsacta.unibo.it/archive/00000437/01/JGM4_Buoli.pdf - A broadband microstrip to waveguide transition for FR4 multilayer PCBs up to 50 GHz.
For those who interested in FR4: from fig 2a,4 and 5 we can see FR4 is used at 28GHz. Maybe some good FR4. Lamda/4 stubs made using wide microstrip stubs (maybe wider band for RF attenuation on DC lines?)

From my experience conventional epoxy resin withfiberglass a.k.a. FR4 is marginal at best for 1GHz let alone 50 due to loss tangent.

Polyamide (GETEK) is much better and only 10% more cost. Teflon (Rogers) or ceramic is best.

Perhaps I am missing something in the slot transformer and thick metal substrate how it makes FR4 almost lossless??

I think losses are low because it is waveguide. There are some papers on SIW made using FR4, they also report results better than microstrips.

I presume the OP knows about the existence of dedicated RF substrates, the loose specification of electrical FR4 parameters and it's relative high dielectric losses at microwave. The losses of FR4 are additionally promoted by the intentional roughness of the copper foils. If FR4 is suitable though for a specific microwave application depends on the specific requirements which haven't been revealed in this case. We should expect that the OP verified the performance of his design, we can't anyway.

I understand that you are asking for a geometry correction during fabrication similar to "controlled impedance" methods to adjust for FR4 variations. Or do you intend tuning of individual boards, e.g. by cutting/milling copper? The term "without equipment" sounds misleading because some kind of test equipment will be used in any case.

You are asking for common practice. In my view it's usual to tune the layout and/or other circuit parameters after prototype fabrication with a specific substrate quality in the lab, then fix this substrate for further production. Preferably use FR4 quality with narrowed specification like Isola FR408.