I made a very simple example to do this using Go, this uses the uplink library as Alexey linked above: share file programatically using storj linksharing service · GitHub
For Android, looks like you can at least create the restricted access grant with this snippet: GitHub - storj/uplink-android: Storj network Android library then post it to https://auth.storjshare.io/v1/access with request body like {"access_grant":"abc123","public":true} to get an access key for Linksharing. The response will look something like {"access_key_id":"abc","secret_key":"abc","endpoint":"https://gateway.storjshare.io"}. Then you can build the Linksharing URL using the access_key_id from that response, as Alexey shows above.