Out of the box? No. Currently there is no SDK or anything like that. It will take some development work to integrate the Adobe Analytics code into your voice app. You could implement with your own team, or have a vendor implement it for you. This link has the documentation on how to implement: Analytics for Digital Assistants
By voice search term, do you mean the exact voice request that is spoken by a user? That is not available, nor would it be from Adobe or any other vendor. You'd be able to collect how the digital assistant interpreted the users voice request, but you won't be able to collect the raw voice request itself.
Let's say a user says, "Alexa, Send John $20 for dinner last night from my banking app." The device (e.g. Amazon Echo Dot) sends the voice request to the digital assistant in the cloud (e.g. Alexa). This digital assistant then parses the voice request into machine-understandable intents and details/parameters. In our example this would be something like Intent = sendMoney; Who = John; Amount = 20; Why = Dinner. The intent and the details are then sent to the voice app, which processes the request.
This means that the full voice request is not available to any system or software beyond the digital assistant. You can definitely collect things like Intent = sendMoney; Who = John; Amount = 20; Why = Dinner, or put them together into something like "sendMoney 20 dollars to John for dinner" (i.e. concatenating the fields with descriptors).
For other examples, if you have a music app, you'll be able to report on what songs/artists/playlists, etc. users are asking the assistant to play.