As suggested in the comments, this question is equivalent to asking for primes of the form 1 +kn, where (n+1) is given and k is an integer at most n+1. The current observations and theory suggest the answer is yes, for every n there is a k less than n+2 so that 1+kn is prime, but we do not have a proof. Linnik's theorem is mentioned in a question link provided in a comment of Lucia, and later results give conditional and unconditional bounds on k. Currently (as far as I know) your question is an open problem.

Much more can be said, however. The question is one of a class of problems of the location of primes in certain arrays. Two such problems are: is there a prime in every row of (a standard square arrangement of) a matrix of numbers from 1 to n^2 for every n > 1? Are there large connected components of primes in an Ulam spiral starting the spiral from any integer n? Your question inspires the following: How many diagonals (columns) contain primes in a (standard n by n) array? Is it always at least $2\phi(n) (\phi(n))$? Here one needs to nail down the definition of diagonal, but that is part of the fun of exploring. Other arrangements of numbers can be considered. As far as I know, hexagonal arrangements have been considered only for Eisenstein primes (cf Guy's UPINT book for more).

Gerhard "Opens Minds With Open Problems" Paseman, 2016.01.16

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