Voice Conversion (VC) emerged as a significant domain of research in the field of speech synthesis in recent years due to its emerging application in voice-assisting technology, automated movie dubbing, and speech-to-singing conversion to name a few. VC basically deals with the conversion of vocal style of one speaker to another speaker while keeping the linguistic contents unchanged. VC task is performed through a three-stage pipeline consisting of speech analysis, speech feature mapping, and speech reconstruction. Nowadays the Generative Adversarial Network (GAN) models are widely in use for speech feature mapping from source to target speaker. In this paper, we propose an adaptive learning-based GAN model called ALGAN-VC for an efficient one-to-one VC of speakers. Our ALGAN-VC framework consists of some approaches to improve the speech quality and voice similarity between source and target speakers. The model incorporates a Dense Residual Network (DRN) like architecture to the generator network for efficient speech feature learning, for source to target speech feature conversion. We also integrate an adaptive learning mechanism to compute the loss function for the proposed model. Moreover, we use a boosted learning rate approach to enhance the learning capability of the proposed model. The model is trained by using both forward and inverse mapping simultaneously for a one-to-one VC. The proposed model is tested on Voice Conversion Challenge (VCC) 2016, 2018, and 2020 datasets as well as on our self-prepared speech dataset, which has been recorded in Indian regional languages and in English. A subjective and objective evaluation of the generated speech samples indicated that the proposed model elegantly performed the voice conversion task by achieving high speaker similarity and adequate speech quality.