You can inherit  crm.view_crm_case_opportunities_filter view  and add the below:

<record id="crm_lead_view_search_notes" model="ir.ui.view">
    <field name="name">crm.lead.view.search.notes</field>
    <field name="model">crm.lead</field>
    <field name="inherit_id" ref="crm.view_crm_case_opportunities_filter"/>
    <field name="arch" type="xml">
          <field name="activity_state" position="after">
                <field name="message_ids" string="Notes" filter_domain="[('message_ids.body','ilike',self)]"/>
            </field>
    </field>
</record>
 

Then you can search in chatter from Search.