Following Devon's advice of using rand() in awk:

gunzip -c my.fq.gz \
| paste - - - - \
| awk 'BEGIN{srand(1234)}{if(rand() < 0.3) print $0}' \
| tr '\t' '\n' \
| gzip > rnd.fq.gz

(Replace 0.3 with the proportion of reads to keep)

However, at a first approximation the reads in a fastq file should be random so something like this might be good enough and possibly faster(?):

...
| awk '{if(NR % 3 == 0) print $0}' \
...

Another option is to use sample with a --lines-per-offset value of 4.

First, get the number of FASTQ records:

$ gunzip -c records.fq.gz | wc -l | awk '{d=$1; print d/4;}'

Multiply the result (say, 12345 records) by the fraction of records you want (say, 42%):

$ calc "floor(12345*42/100)"
5184

Then sample:

$ gunzip -c records.fq.gz > records.fq
$ sample --lines-per-offset=4 --sample-size=5184 records.fq > sample.fq