Coronavirus is an epidemic that greatly distresses people all over the globe. The similarities amongst the COVID-19 and other lung diseases are highly equivalent and it makes it difficult to diagnose it. The currently presented deep learning (DL) models offer a method to identify the presence of COVID-19 from the radiological images. In this view, this paper presents a new deep convolutional neural network based AlexNet model with Gradient Boosting Tree (DCNNAN-GBT) model for COVID-19 diagnosis and classification. The presented DCNNAN-GBT model involves Gaussian filtering (GF) based preprocessing approach to remove the noise exists in the radiological images. In addition, AlexNet model is employed as a feature extractor to derive a useful set of feature vectors. Besides, GBT model is applied as a classification technique to allocate proper class labels to the input images. A wider experimental analysis was performed to highlight the effective COVID-19 diagnostic outcome. The experimental results pointed out that the DCNNAN-GBT model has resulted in a maximum average sensitivity of 94.32%, specificity of 93.10%, and accuracy of 94.13%.