Can we able to measure the gain of any test antenna on VNA with out using standard gain measurement set up procedure (two antennas: one is AUT and another one is horn antenna).
Yes, if you have two similar test antennas just apply Friis formula to them in a lab link- the gain will be squared since they're the same, all other terms will be known, you'll get their gains.
Pr=(Pt.G^2)/(4.pi.d/lambda)^2 This is friis. G is the gain of the antennas (two similar ones, with the same gain). You place them at a known distance d (should be farfield), excite one of them with a power source of known frequency (lambda) and power (the Pt) and measure on the other end with a spectrum analyzer the received power.
You can take a look into Balanis Book, Antenna Theory, on the equation, chapter 2.
Yes it is possible to measure the gain of any test antenna by itself on VNA without the need for a second antenna. The method is called as gain measurements using single-antenna method. The test antenna alone is used but with a PEC sheet. The sheet creates a second image antenna. You can check in google scholar by using keywords such as "single antenna" method of gain measurement.
Example paper:
Felic, Gordana. "Antenna gain measurements in the V-band: A single-antenna method." 2009 Electromagnetic Compatibility Symposium Adelaide. IEEE, 2009.