As long as your input file is not too large you could slurp the whole file into one string and do global matches against it. Use the s flag in the regex to allow the dot metacharacter to match newlines.

use strict;
use warnings;

open my $inputFH, q{<}, \ <<'END_OF_FILE' or die qq{open: $!\n};
end
sleep 10
dis qremote(MQSI.3PL846)RNAMERQMNAME1 : dis qremote(MQSI.3PL846) RNAME
+ RQMNAME
AMQ8409: Display Queue details.QUEUE(MQSI.3PL846)TYPE(QREMOTE)RQMNAME(
+MSTBKRQ1)RNAME(MQSI.3PL846)
end2 : end

Starting MQSC for queue manager NTTCSWQ1.

AMQ8409: Display Queue details.QUEUE(MQSI.3PL944)TYPE(QREMOTE)RQMNAME(
+MSTBKRQ1)RNAME(MQSI.3PL944)
exit
5724-H72 (C) Copyright IBM Corp. 1994, 2004.  ALL RIGHTS RESERVED.
     1 : dis qremote(MQSI.ADM850) RNAME RQMNAME
AMQ8409: Display Queue details.
   QUEUE(MQSI.ADM850)                      TYPE(QREMOTE)
   RQMNAME(MSTBKRQ1)                       RNAME(MQSI.ADM850)
     2 : end
end
sleep 10
exit
AMQ8409: Display Queue details.
   QUEUE(MQSI.ADMAPTR)                     TYPE(QREMOTE)
   RQMNAME(MSTBKRQ1)                       RNAME(MQSI.ADMAPTR)
     2 : end
One MQSC command read.
No commands have a syntax error.
All valid MQSC commands were processed.
END_OF_FILE

my $lines = do
   {
       local $/;
       <$inputFH>;
   };

close $inputFH or die qq{close: $!\n};

my $rxExtract = qr
   {(?xs)
       AMQ8409
       .*?
       QUEUE\(([^)]+)\)
       .*?
       RQMNAME\(([^)]+)\)
       .*?
       RNAME\(([^)]+)\)
   };

while ( $lines =~ m{$rxExtract}g )
{
    print qq{QUEUE: $1, RQMNAME: $2, RNAME: $3\n};
}
[download]

The output.

QUEUE: MQSI.3PL846, RQMNAME: MSTBKRQ1, RNAME: MQSI.3PL846
QUEUE: MQSI.3PL944, RQMNAME: MSTBKRQ1, RNAME: MQSI.3PL944
QUEUE: MQSI.ADM850, RQMNAME: MSTBKRQ1, RNAME: MQSI.ADM850
QUEUE: MQSI.ADMAPTR, RQMNAME: MSTBKRQ1, RNAME: MQSI.ADMAPTR
[download]

I hope this is helpful.

Cheers,

JohnGG