Let's make some assumptions before writing code:
- Board Name same as project name
- There will be only one board with that name
- We have a Sprint model
I am using Jersey client here to retrieve data from JIRA.
private Client jerseyClient = Client.create();
jerseyClient.addFilter(new HTTPBasicAuthFilter("username", "password"));
private Gson gson = new Gson();
Helper methods
   /**
     * This method will a GET request to the URL supplied
     * @param url to request
     * @return String response from the GET request
     */
    public String makeGetRequest(String url){
        ClientResponse response = jerseyClient.resource(url).accept("application/json").get(ClientResponse.class);
        return response.getEntity(String.class);
    }
    /**
     * This method helps in extracting an array from a JSON string
     * @param response from which Array need to be extracted
     * @param arrayName
     * @return JsonArray extracted Array
     */
    public JsonArray extractArrayFromResponse(String response, String arrayName){
        JsonObject jsonObject = gson.fromJson(response, JsonObject.class);
        return jsonObject.get(arrayName).getAsJsonArray();
    }
Code for retrieving sprints
    /**
     * This method will retrieve list of sprints in a given project
     * @param project for which we are requesting sprints 
     * @return List of sprints
     */
     public List<Sprint> getSprints(String project) {
        List<Sprint> sprintList = new ArrayList<>();
        try {
            //get board URL for the given 
            String boardUrl = "https://jira.company.com/rest/agile/1.0/board?name=" + URLEncoder.encode(project, "UTF-8");  //assumption 1
            String boardResponse = makeGetRequest(boardUrl);
            JsonArray boards = extractArrayFromResponse(boardResponse, "values");
            if(boards.size() > 0){
               JsonObject board = boards.get(0).getAsJsonObject(); //assumption 2
            //get all sprints for above obtained board
            String sprintUrl = jsonHandler.extractString(board, "self")+"/sprint";
            String sprintsResponse = makeGetRequest(sprintUrl);
            JsonArray sprints = extractArrayFromResponse(sprintsResponse, "values");
                //loop through all sprints
                for (int i = 0; i < sprints.size(); i++) {
                    JsonElement sprint = sprints.get(i);
                    JsonObject sprintObj = sprint.getAsJsonObject();
                    String sprintName = sprintObj.get("name").getAsString();
                    Sprint sprint = new Sprint(sprintName);
                    sprintList.add(sprint);
                }//end of for loop
            }
        } catch (UnsupportedEncodingException e) {
            e.printStackTrace();
        }
        return sprintList;
    }