Is there a more elegant solution to the bwa-mem: paired reads have different names error?
Assuming the read name lines only contain things like @JQ714243-920-1 and nothing else, then sed -i '1~4 s/-[12]$//' file.fastq (N.B., example updated a bit for safety, thanks to @terdon) will remove the -1 and -2 suffixes.

However, you should try to figure out why the read names were munged into that form to begin with (at least Illumina machines won't produce that, I don't know about others).