There's nothing stopping you from stripping the quality lines from a fastq and turning it into a fasta, then running the sequences in blast. Not sure what'd you get out of it asides from a nearest match for each read to a member of your index. Are you trying to count something, map something or assign identities before a de novo assembly? Mapping the reads to the fasta that your nr is derived from might give you visualization of your reads in context of your database if that is what you were looking for.

If your insert length is short enough and your read length long enough, merging reads may give you more to work with; and deduplication may eliminate some reads that correspond to overly represented areas that might save you some runtime.