If you're searching for hits within a larger text, you don't want to use ^ and $ as some other responders have said; those match the beginning and end of the text. Try this instead:
\bdbo\.\w+_fn\b
\b is a word boundary: it matches a position that is either preceded by a word character and not followed by one, or followed by a word character and not preceded by one. This regex will find what you're looking for in any of these strings:
dbo.functionName_fn
foo dbo.functionName_fn bar
(dbo.functionName_fn)
...but not in this one:
foodbo.functionName_fnbar
\w+ matches one or more "word characters" (letters, digits, or _). If you need something more inclusive, you can try \S+ (one or more non-whitespace characters) or .+? (one or more of any characters except linefeeds, non-greedily). The non-greedy +? prevents it from accidentally matching something like dbo.func1_fn dbo.func2_fn as if it were just one hit.