I think you can do it by setting the --info-file option and then parsing the resulting file to get the reads of interest, see info file format. To get fastq format from the info-file should be straightforward. If I understand your question correctly, from the info file you want columns:

1 Read name

6 Sequence of the read that was matched to the adapter

10 Quality values corresponding to sequence matched to the adapter (can be empty)

(Make sure you don't need to reverse complement)

With BBTools, you can do this:

bbduk.sh in=reads.fastq out=unmatched.fastq outm=matched.fastq literal=GTNNNNTG copyundefined k=8 mm=f

However, your motif is not specific enough to be useful. It matched 77.05% of read pairs in some random data I'm playing with that have nothing to do with your adapters. That said, if you know that, for example, they only occur in the first 8bp of read 1, you could add "restrictleft=8 skipr2", which cuts it down to a 0.65% false-positive rate.