Find the Emirps!

05AB1E, 17 bytes

Uses CP-1252 encoding.

Input order is n, b
Output is in base-10.

µN²BÂD²öpŠÊNpPD–½

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Explanation

                    # implicit input a,b
µ                   # loop until counter is a
 N²B                # convert current iteration number to base b
    ÂD              # create 2 reversed copies
      ²ö            # convert one reversed copy to base 10
        p           # check for primality
         ŠÊ         # compare the normal and reversed number in base b for inequality
           Np       # check current iteration number for primality
             P      # product of all
              D     # duplicate
               –    # if 1, print current iteration number
                ½   # if 1, increase counter

Jelly, 16 bytes

bµU,ḅ⁹QÆPḄ=3
⁸ç#

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How?

bµU,ḅ⁹QÆPḄ=3 - Link 1, in-sequence test: n, b
b            - convert n to base b - a list
 µ           - monadic chain separation
  U          - reverse the list
   ,         - pair with the list
     ⁹       - link's right argument, b
    ḅ        - convert each of the two lists from base b
      Q      - get unique values (if palindromic a list of only one item)
       ÆP    - test if prime(s) - 1 if prime, 0 if not
         Ḅ   - convert to binary
          =3 - equal to 3? (i.e. [reverse is prime, forward is prime]=[1,1])

⁸ç# - Main link: b, N
  # - count up from b *see note, and find the first N matches (n=b, n=b+1, ...) for:
 ç  - last link (1) as a dyad with left argument n and right argument
⁸   - left argument, b

* Note b in base b is [1,0], which when reversed is [0,1] which is 1, which is not prime; anything less than b is one digit in base b and hence palindromic.

Mathematica, 70 bytes

Cases[Prime@Range@437,p_/;(r=p~IntegerReverse~#2)!=p&&PrimeQ@r]~Take~#&

Works for 0 <= n <= 100 and 2 <= b <= 16. From the list Prime@Range@437 of the first 437 primes, find the Cases pwhere the IntegerReverse r of p in base #2 is not equal to p and is also prime, then take the first # such p.

Here's a 95 byte solution that works for arbitrary n>=0 and b>=2:

(For[i=1;a={},Length@a<#,If[(r=IntegerReverse[p=Prime@i,#2])!=p&&PrimeQ@r,a~AppendTo~p],i++];a)&