No.
HAL covers only a small subset of what the STM32 hardware can do. It is good enough to get started with STM32, create a prototype fast, but its limitations become soon apparent.
As far as I can tell, the UART idle interrupt is neither checked nor handled in the HAL interrupt handler. Enabling, checking and handling it separately might or might not confuse HAL code.
You can of course set
USARTx->CR1 |= USART_CR1_IDLEIE;
in your code, and have an interrupt handler which can deal with it. Then you must either check the HAL code that it won't get confused by anything you do with the registers, or implement all of the UART handling without HAL. The latter approach has the advantage that the register interface is far better documented as HAL functions.
DMA TC and HT interrupts are handled by HAL, but they are not exactly useful for this task (maybe TC for detecting buffer rollover). You can read the DMA stream NDTR register to find out the amount of data in the buffer.
You might also want to check out the UART receiver timeout feature (RTOR register and the associated interrupt mask and flag bits) which provides a more flexible way of detecting gaps in the transmission.