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When dealing with practical problems of stress-strength reliability, one can work with fatigue life data and make use of the well-known relation between stress and cycles until failure. For some materials, this kind of data can involve extremely large values. In this context, this paper discusses the problem of estimating the reliability index R = P( Y < X) for stress-strength reliability, where stress Y and strength X are independent q-exponential random variables. This choice is based on the q-exponential distribution's capability to model data with extremely large values. We develop the maximum likelihood estimator for the index R and analyze its behavior by means of simulated experiments. Moreover, confidence intervals are developed based on parametric and nonparametric bootstrap. The proposed approach is applied to two case studies involving experimental data: The first one is related to the analysis of high-cycle fatigue of ductile cast iron, whereas the second one evaluates the specimen size effects on gigacycle fatigue properties of high-strength steel. The adequacy of the q-exponential distribution for both case studies and the point and interval estimates based on maximum likelihood estimator of the index R are provided. A comparison between the q-exponential and both Weibull and exponential distributions shows that the q-exponential distribution presents better results for fitting both stress and strength experimental data as well as for the estimated R index. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]