The way Bitcoin works is that coins which have not yet been spent (Unspent Transaction Outputs or UTXO’s) are linked to a specific public key. This public key is in turn linked to the private key of the address.
The private key can be compared to a password and proves ‘ownership’ of certain addresses. Without this private key it is not possible to send any coins from the address since you are unable to prove your ownership of the address. As Bitcoin evangelist Andreas Antonopoulos said: “If you don’t own your private key, you don’t own your bitcoins.”
In this case it is not possible to convert your non spendable BTC to spendable BTC without the private key. We would therefore recommend that you search for a backup of the private key.