If Blockchain miners weren't paid to mine would there still be trust in all bitcoin transactions?
Without a sufficient number of miners, security of the bitcoin network would degrade. If an attacker (or pool of attackers) could come close to amassing enough computing power to match the computing power of the rest of the miners, the attacker could disrupt the network by deleting newly-written blocks of transactions from the blockchain, preventing certain transactions from being written to blocks in the blockchain, and/or double-spending bitcoins.
Miners are compensated in two ways:
1) Each time a block is successfully mined, the miner of that block is awarded newly-minted bitcoins. At the moment, the amount awarded is 25 bitcoins.
2) Miners collect transaction fees for the transactions in the blocks that they mine. The amounts of these transaction fees are determined by the payers of the transactions.
Currently, the transactions fees that the miners collect are usually nominal compared to the amount that miners earn in newly-minted bitcoins for the blocks that they mine. But, that could change. As mentioned above, the amount of newly-minted bitcoins that is awarded for a mined block is currently 25 BTC, but this amount is halved after every 210,000 blocks mined. At this rate, it is projected to halve to 12.5 BTC sometime in 2016, then halve again to 6.25 BTC sometime in 2020.
As the reward in newly-minted bitcoins decreases, there is speculation that fewer miners will find it viable to continue to mine. The bitcoin network relies in miners to keep it secure, so it may come to a point where payers may be inclined to pay more in transaction fees to incentivize the miners to mine their transactions.
Trust would remain in Bitcoin transactions only if miners continued to mine. However, without a financial incentive, there would be no reason for them to shoulder the high electricity costs of mining.