I am sure that no conversion of photon energy is happening here. 

Especially you need an up-conversion in energy which is very unlikely. Normal flourescence cannot be the cause here. There are detector cards for up-conversion of laser light, but they need to be "charged" by sunlight before they can be used . And this is very special material.


More likely the laser is relatively strong and the sensitivity of the eye is still sufficiently high.

For example: While I was working with 762 nm laser (Oxygen A-band) I, and all my colleagues, were able to clearly see the beam (even though 762 nm is already classified as IR). The laser had a power of ~ 300 µW and the collimated beam was clearly visible on a sheet of white paper at daylight conditions. When spread over 1 cm area the beam was very visible with light of.

Even though 780 nm is certainly further in the IR than 760 nm, your laser might be more powerful and the eye is still able to see the beam. 

But the beam will probably be much more powerful than the perceived brightness suggests.