With awk like so:

awk '{for (i=1; i<=NF; ++i) {if ($i ~ "TRINITY_") {$0=">"$i}}}1' file.fasta

If TRINITY_ is found in a field then replace the whole line with that field preceded with > and print all lines ... The above command will not edit the original file file.fasta but, will just output the lines ... To get the output in a file like output.fasta instead, do it like so:

awk '{for (i=1; i<=NF; ++i) {if ($i ~ "TRINITY_") {$0=">"$i}}}1' file.fasta > output.fasta

or edit the original file in-place(the original file will be modified) with gawk like so:

gawk -i inplace '{for (i=1; i<=NF; ++i) {if ($i ~ "TRINITY_") {$0=">"$i}}}1' file.fasta