No, this service is only provided in Bali.
In the Gili Islands, there is no motorised transport, but the islands are small. Travel is by horse and cart (cidomo), which you can arrange upon arrival if your hotel is a long walk away, or contact your hotel beforehand to have it standing by when you arrive.
In Lombok, we strongly advise that you ask your hotel to arrange a pick-up from the harbour at Teluk Kode – they all know where to go. The local community does not allow taxis, Uber, Grab etc to wait at our harbour area, and you will not get the best prices from transport providers on standby. However, you can take local taxis from within Lombok (eg from Senggigi) TO the port area. Taxis can drop you off at the harbour, but cannot pick you up.