The rapid development of Industrial IoT (IIoT) opens up promising possibilities for data analysis and machine learning in IIoT networks. As a distributed paradigm, federated learning (FL) allows numerous IIoT devices to collaboratively train a global model without collecting their local data together in central servers. Unfortunately, a centralized server used to aggregate local gradients can be compromised and forge the result, which incurs the need for aggregation verification. Several approaches focusing on verifying the correctness of aggregation have been proposed. However, it is still an open problem since devices have to devote more computation resources for verification, which are especially not friendly to resource-constrained IIoT devices. Furthermore, verifying weighted aggregation has not been supported in existing approaches. In this paper, we propose an efficient verifiable federated learning approach for IIoT networks, which verifies the aggregation of gradients and requires lowest burden on IIoT devices by introducing zero-knowledge proof techniques. Moreover, our design supports weighted aggregation verification to validate the aggregation of weighted gradients in the cloud server. By comparing the proposed approach with the state-of-the-art schemes including VerifyNet and VeriFL, we demonstrate the superior performance of our approach for resource-constrained devices, which minimizes the computational overheads of the IIoT devices.