class MainForm(FlaskForm):
    select_val = SelectField('Выбор', choices = [('FIRST', 'FIRST'), ('SECOND', 'SECOND')])
    main_name = StringField('Имя')
    submit = SubmitField('Выбрать')

@app.route('/result')
def result():
    return render_template('result.html', main_name=session.get('main_name'))

@app.route('/select_page', methods=['GET','POST'])
def select_page():
    form = MainForm()
    if form.validate_on_submit():
        select_val = form.select_val.data
        session['main_name'] = form.main_name.data
        return redirect(url_for('result'))
    return render_template('select_page.html', form=form)

@app.route('/select_pageprocess', methods=['POST'])
def select_pageprocess():
    select_val = request.form['select_val']
    return jsonify({'select_val' : select_val})

$(document).ready(function() {
$('select').on('change', function(event) {
    $.ajax({
        data : {
            select_val : $('#select_val').val()
        },
        type : 'POST',
        url : '/select_pageprocess'
    })
    .done(function(data) {
        if(data.select_val == 'FIRST'){
            $('#second').hide();
            $('#first').show();

        }else if(data.select_val == 'SECOND'){
            $('#first').hide();
            $('#second').show();
        }
    });
    event.preventDefault();
});
});

<form action="" method="post" id="select_pageform" novalidate>
    {{ form.hidden_tag() }}
    <p>
        {{ form.select_val.label }}<br>
        {{ form.select_val }}
        {% for error in form.select_val.errors %}
        <span style="color: red;">[{{ error }}]</span>
        {% endfor %}
    </p>
    <p>{{ form.submit() }}</p>
</form>
<div id="first" style="display:yes;>"{% include '_FIRST.html' %}</div>
<div id="second" style="display:none;>"{% include '_SECOND.html' %}</div>


<p>first</p>
{{ form.main_name.label}}<br>
{{ form.main_name}}<br>

<p>second</p>
{{ form.main_name.label}}
{{ form.main_name}}