Understanding the progress of the Sustainable Development Goals (SDGs) proposed by the United Nations (UN) is important, but difficult. In particular, policymakers would need to understand the sentiment within the public regarding challenges associated with climate change. With this in mind and the rise of social media, this work focuses on the task of uncovering the sentiment of Twitter users concerning climate-related issues. This is done by applying modern natural-language-processing (NLP) methods, i.e. VADER, TextBlob, and BERT, to estimate the sentiment of a gathered dataset based on climate-change keywords. A transfer-learning-based model applied to a pre-trained BERT model for embedding and tokenizing with logistic regression for sentiment classification outperformed the rule-based methods VADER and TextBlob; based on our analysis, the proposed approach led to the highest accuracy: 69%. The collected data contained significant noise, especially from the keyword 'energy'. Consequently, using more specific keywords would improve the results. The use of other methods, like BERTweet, would also increase the accuracy of the model. The overall sentiment in the analyzed data was positive. The distribution of the positive, neutral, and negative sentiments was very similar in the different SDGs.