Your filename is a relative reference, where system\ is a subfolder of the same folder where your mirc.ini is located.
take a look at the s/w switches for $read. It depends on the format for your line. If the first word of each line is a nick and you want to display only the text following that first word:
$read(system\birthday.txt,nts,$nick)
This searches for a delimited word, so searching for the nick TEST won't find the line TEST1. It's equivalent to $gettok($read(system\birthday.txt,ntw,$nick *),2-,32)
If you want to find a line where $nick is a space delimited word in the line and return that entire line, your search should include the spaces: $read(system\birthday.txt,ntw,* $nick *)