Can I Call Upon Allah by the Name “Ya Hayyu Ya Qayuum?”

Can I call upon Allah by the name “ya hayyu ya qayuum?”
Dear questioner,
May Allah bless you and increase you in good.
It is permissible to call upon Allah Most High by saying, “Ya Hayyu Ya Qayyum.” These are two of His blessed names- al Hayyu and al Qayyum.
Sunna Supplications 
Anas bin Malik said: “Whenever a matter would distress him, the Prophet (Allah bless him and give him peace) would say: “O Living, O Self-Sustaining Sustainer! In Your Mercy do I seek relief.” [Tirmidhi]
From this hadith of Anas, we can see that the beloved Prophet (Allah bless him and give him peace) himself used to call upon Allah by these names.
Divine Names 
There are many virtues and merits by calling upon Allah by the divine names and its different forms. Consider just one narration that includes these two very names you had asked about;
[عن أنس بن مالك:] كنتُ جالِسًا مع النبِيِّ – ﷺ – في المسجدِ، ورَجُلٌ يُصلِّي، فقال: اللَّهُمَّ ! إني أسْألُكَ بأنَّ لك الحمدُ، لا إله إلا أنتَ الحنّانُ المنّانُ، بديعُ السماواتِ والأرضِ، يا ذا الجلاَلِ والإكْرامِ ! يا حَيُّ يا قَيُّومُ ! أسْألُكَ، فقال النبِيُّ – ﷺ -: دعا اللهَ باسْمِهِ الأعْظَمِ؛ الذي إذا دُعِيَ به أجابَ، وإذا سُئِلَ به أَعْطى
“I was sitting with the Messenger of Allah (Allah bless him and give him peace), and a man was standing and praying. When he bowed, prostrated, and recited the tashahhud, he supplicated, and in his supplication he said:
“O Allah, indeed I ask of You since all praise is due to You — there is none worthy of worship but You, The Bestower, The Creator of the heavens and earth, O Possessor of majesty and honor, O EVER-LIVING, O ETERNAL! I ask of You.”
The Prophet (Allah bless him and give him peace) said: “‘Do you know what he has supplicated with?’ They said: ‘Allah and His Messenger know best.’ He said: ‘By the One in Whose Hand is my soul, he called upon Allah by His Greatest Name, which if He is called by it, responds, and if He is asked by it, He gives.’” [Nisa’i and Abu Dawud]
Etiquette of Dhikr 
I highly recommend that you have a quick look at the following article:
May Allah bless you and keep you in His remembrance.
[Imam] Yama Niazi
Checked and Approved by Shaykh Faraz Rabbani