This is interesting, because I don't feel the film showed that. However, what it did show was - in the case of Spock and Kirk, their journey to respecting one another as individuals and learning how to blend their particular abilities into an early relationship that will ultimately become the friendship they later had.
McCoy is also shown to play a part in this, but quite correctly, is always closer to Kirk than Spock. This was true of TOS, and so it will take a bit of time for that actual friendship to develop.
'If the Apocalypse starts, beep me!' - Buffy Summers
'The sky's the limit.....' Jean-Luc Picard, 'All Good Things'
courtesy of Saquist