Would the engineer team here on Earth detect it? How?
There are two issues here.
Suppose Voyager 2 (Voyager 1 is moving faster and is further away, so the effect will be lesser on Voyager 1 than on Voyager 2) made a ridiculously close flyby of an object with a similar size and mass of Neptune. Such a flyby would result in a delta-V of about 8.2 km/sec. Suppose this delta-V is orthogonal to the line of sight to Earth (worst case). Even then, it would be almost 41 days to fly out of the beam width of the Deep Space Network antennae. To this day, JPL still receives data from the Voyagers on a daily basis. That gives plenty of time to notice the change in velocity. It appears that continued contact with the Voyagers could continue even in the face of a ridiculously close flyby of a ridiculously large object.
However, such a ridiculously close flyby would have another effect. We don't "see" the Voyager satellites. We instead receive their transmissions. The vehicles are invisible if we can't receive those transmissions.
The problem is that a flyby not only changes the satellite's velocity, it also changes the satellite's orientation and rotation rate thanks to gravity gradient torque. That's fine in the case of a planned flyby. It's no so fine in the case of an unplanned flyby when the vehicle has limited fuel to correct for an unplanned rotation and limited intelligence to orient itself so the antenna points back to Earth.
The only downlink capability currently available to Voyagers is the X-band transmission, with a beam width of 0.5 degrees. Even a small unplanned change in Voyager attitude means we on Earth couldn't see that Voyager, at least not until the Voyager has found out where it is pointing and corrected for that. A change in attitude rate is an even bigger problem. The Voyagers have a very limited amount of remaining fuel for attitude control. While that limited amount of fuel is more than enough to last until 2025 assuming that the vehicles are flying through empty space, whether it's enough to correct for an unexpected large gravity gradient torque is a very different matter.