Importing Ticket History from Previous Application

Can you export the ticketing from another Service Desk Application and import into HubSpot as ticket history & assign to contacts?

You can import tickets, you just want to make sure you have that information in a CSV XLSX or XLS file format and you would want the column labels to match your ticket properties in HubSpot. You also need to make sure that the ticket status matches the categories within your ticket pipeline in HubSpot. 
To associate with a contact you would need to create a column in the spreadsheet for (HubSpot) Customer ID and match in the spreadsheet before importing. 
Notes: You would only be able to match contacts that are already in HubSpot. If you had contacts outside of that you would have to upload the customers first and then export with the HubSpot ID included. 
Alternatively - you can Import multiple objects and select Contacts and Tickets. 
You would need one file for your tickets and one for your contacts and they would need a matching property between them this works best with a record ID, but since none of them would likely have a HubSpot unique ID at this point it may be worth creating a custom property for both record types and using either the IDs from your previous system or your own ID.