No, they generally don't, although if a precinct is in backlog (too many pending radio runs, not enough available units) they can lend a hand.
A patrol borough task force (in this case Staten Island), is a group of officers who's primary function is to answer mobilizations. Mobilizations are called when an event occurs within a precinct that taxes that precinct's available resources. One example of when a mobilization may get called is for a search for a missing child. While one precinct sector may be able to handle the preperation of the missing person report, it can not effectively conduct a search of an area to look for this kid. So the precinct patrol supervisor can request a Level 1 Mobilization to bring in the appropriate task force (approx. eight+ more officers). Once there, those officers can now conduct a seach of building hallways, yards, alleyways, surrounding buildings, etc., etc., etc. Other examples of when Level 1s might get called is for shooting (tends to draw large crowds at crime scenes - and ups the opportunity for retailitory gun play - thus flood the area with cops to help quiet things back down....)
While not operating at mobilizations, patrol borough task forces generally rove within certain precincts within their borough or perform select enforcement functions (i.e. DWI, radar).
Should an event be big enough that a borough task force needs additional support, a Level 2 Mobilization can be called. This will bring in another task force from an adjoining patrol borough (or all of them, depending).
There is an option called a Level 3 Mobilization. This will bring the appropriate patrol borough task force (a Level 1) plus at least one car from each precinct within that patrol borough. So, sticking with your example, that would bring to an incident the SITF and one car each from the 120, 122, and 123 PCTs. This is a quicker way to get additional help to a scene, as opposed to waiting for it to come from over a bridge (i.e. Brooklyn South TF into SI or Bronx TF into Queens North).
A Level 4 Mobilization would bring the combined equivalent of a Level 2 (all PB TFs) and at least one car from all of the precincts citywide.
Note that from Levels 2 through 4 assorted additional specialized equipment (a mobile command post, an Emergency Service unit, at least one helicopter, Mounted horses, etc., etc.) will be automatically assigned and can be cancelled or supplemented as needed.