Are there public $p$ and $q$ numbers for use in DSA?
There is a very detailed algorithm on how to construct $p,q$ in FIPS 186-3, appendix A.1.2.1.2. The algorithm provided takes as input, the lengths $L,N$, of $p,q$ (resp.) (for instance L=1024,N=160) and output, the primes $p,q$ such that $p\equiv 1\pmod{q}.$
Also,there is an implementations in python (for $(L,N)=(1024,160)$). 
For instance

from Crypto.PublicKey import DSA
import random
def randfunc(n):
    return ''.join(str(random.random())[4] for _ in xrange(n))
DSAkey=DSA.generate(int(1024),randfunc)
p=DSAkey.p;q=DSAkey.q; 


Running the previous I got

p=898846567431157967424297114057633644601771516927834298008846524493109792\
637522535293491954598238817151457964980464592383454281215613866269456797\
539564000773528820716639254597505008070182540287714904340213156913571237\
346370468948761234961687162517352526627424620993348024330584723776744085\
98573487858308054417L
q= 1193447034984784682329306571139467195163334221569L