The GPS module generally uses serial communication. Presumably the Trimble is such a device. TheATmega328 has only one hardware UART which is usually dedicated to the USB interface.
You need to use a software serial (library) function to attach a second serial device such as a GPS - but it is doable.
A GPS antenna alone won't do anything! A GPS module on the otherhand communicates
by a serial line so can be connected to any Arduino easily. However you do need to
make sure the right supply voltage is provided to the GPS, if its 3.3V you will need
a 3.3V regulator (the on-board 3.3V isn't powerful enough and also doesn't work without
USB, which sounds like what you are worried about).
The GPS antenna connects directly to the GPS module somehow. The GPS module also has a connection to the Arduino ( usually a serial connection ). The GPS antenna does not connect directly to any part of the Arduino.
The GPS modules that you can buy for around $30, there are several different brands.
The antenna for these is a chunky square thing about 18 mm square and about 8 mm thick. They are usually a pink color for some reason that is unknown to me. For most GPS modules, the antenna is glued directly on top of the GPS microchip. Some of them have a cable connection so you can put the antenna somewhere else.
The GPS module connects to the Arduino by a serial connection, normally 4 wires, +5V, ground, RX and TX. Some of them have the Rx and Tx incorrectly labelled.
Using a USB to serial converter, you can connect the GPS module directly to your computer and use one of several free applications to interact with it.
The antenna itself is of no use to you. You cannot connect the antenna to an Arduino. The Arduino has no RF processing capability. The antenna is only useful when connected to a GPS module.
Amongst my collection - here somewhere - is a GPS unit - complete, looks rather like the antenna only as nowadays they are usually integrated into one unit - which is on the end of a lead with a USB plug on the other end.
It sounds as if one of those would suit you - if the interface is documented and reasonably straightforward - connected directly to the PC/ laptop and with suitable software.
A GPS receiver, commonly called a GPS module these days (as they are now tiny), is an
extremely sophisticated piece of hardware that includes dozens of extremely sensitive
receiver circuits, significant compute power and highly precise hardware timers that
perform trilateration on the demodulated satellite signals to ns accuracy - without such a unit
a GPS antenna is entirely useless! Such units also incorporate significant ROM memory
holding a map that corrects for the lumpiness of the earth's geoid so that altitiude
readings are accurate.
The square antennas are a slab of ceramic dielectric material that act as resonant
devices at the GPS signal frequency, normally called a "patch antenna".